There is now a new recording app called AirShou (link at bottom of post). It works by tricking your device into AirPlaying and Mirroring itself and then recording that. The FPS is quite smooth and I did not seem to have any issues recording or sharing with others.

After downloading the application, you will need to go to Settings -> General -> Profiles & Device Management -> Wise Technologies -> click Trust -> then open AirShou. After this, you just follow the simple steps to record!